1. Tardigrade
  2. Question
  3. Physics
  4. Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II: Column I Column II A The object moves on the x-axis under a conservative force in such a way that its speed and position satisfy v =c1 √c2-x2, where c1, and c2 are positive constants. P The object executes a simple harmonic motion. B The object moves on the x-axis in such a way that its velocity and its displacement from the origin satisfy v=-k x, where k is a positive constant. Q The object does not change its direction. C The object is attached to one end of a mass-less spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration. R The kinetic energy of the object keeps on decreasing. D The object is projected from the earth's surface vertically upwards with a speed 2 √G Me / Re, where Me is the mass of the Earth and Re is the radius of the Earth. Neglect forces from objects other than the Earth. S The object can change its direction only once.

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Q. Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II:
Column I Column II
A The object moves on the $x$-axis under a conservative force in such a way that its "speed" and "position" satisfy $v =c_{1} \sqrt{c_{2}-x^{2}}$, where $c_{1}$, and $c_{2}$ are positive constants. P The object executes a simple harmonic motion.
B The object moves on the $x$-axis in such a way that its velocity and its displacement from the origin satisfy $v=-k x$, where $k$ is a positive constant. Q The object does not change its direction.
C The object is attached to one end of a mass-less spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration. R The kinetic energy of the object keeps on decreasing.
D The object is projected from the earth's surface vertically upwards with a speed $2 \sqrt{G M_{e} / R_{e}}$, where $M_{e}$ is the mass of the Earth and $R_{e}$ is the radius of the Earth. Neglect forces from objects other than the Earth. S The object can change its direction only once.

JEE AdvancedJEE Advanced 2007

Solution:

$( A ) \rightarrow( P )$
In SHM, $v=\omega \sqrt{a^{2}-x^{2}}$ which resembles
$v=c_{1} \sqrt{c_{2}-x^{2}}$
and hence the object executes simple harmonic motion.
(B) $\rightarrow(Q),(R)$
We have
$v =- kx =\frac{d x}{d t} \Rightarrow \int \frac{d x}{x}=- k \int dt$
That is,
In $(x)=-k t \Rightarrow x=e^{-k t}$
Therefore,
$v =\frac{d x}{d t}=-k e^{-k t}$
Now,
K.E. $=\frac{1}{2} m v^{2}=\frac{1}{2} m k^{2} e^{-2 k t}$
That is, the object does not change its direction as shown in the following graph:
image
The kinetic energy keeps on decreasing as shown in the following graph:
image
$( C ) \rightarrow( P )$
We have $T=2 \pi \sqrt{\frac{m}{k}} ;$
therefore, the motion of the object is SHM.
(D) $\rightarrow( Q ),( R )$
As the object goes up against gravity, its speed and hence the kinetic energy goes on decreasing. Also, since
$V =2 \sqrt{\frac{G M_{e}}{R_{e}}}>$ Escape speed $=\sqrt{\frac{2 G M_{e}}{R_{e}}}$
we conclude that the object cannot return to Earth once again.